c 2000 Society for Industrial and Applied Mathematics

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1 SIAM J. OPIM. Vol. 10, No. 3, pp c 2000 Society for Industrial and Applied Mathematics CONES OF MARICES AND SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES MASAKAZU KOJIMA AND LEVEN UNÇEL Abstract. Let F be a compact subset of the n-dimensional Euclidean space R n represented by (finitely or infinitely many) quadratic inequalities. We propose two methods, one based on successive semidefinite programming (SDP) relaxations and the other on successive linear programming (LP) relaxations. Each of our methods generates a sequence of compact convex subsets C k (k = 1, 2,... ) of R n such that (a) the convex hull of F C k+1 C k (monotonicity), (b) k=1 C k = the convex hull of F (asymptotic convergence). Our methods are extensions of the corresponding Lovász Schrijver lift-and-project procedures with the use of SDP or LP relaxation applied to general quadratic optimization problems (QOPs) with infinitely many quadratic inequality constraints. Utilizing descriptions of sets based on cones of matrices and their duals, we establish the exact equivalence of the SDP relaxation and the semiinfinite convex QOP relaxation proposed originally by Fujie and Kojima. Using this equivalence, we investigate some fundamental features of the two methods including (a) and (b) above. Key words. semidefinite programming, nonconvex quadratic optimization problem, linear matrix inequality, bilinear matrix inequality, semi-infinite programming, global optimization AMS subject classifications. 15A48, 52A47, 49M39, 90C05, 90C25, 90C26, 90C30, 90C34 PII. S Introduction. Consider a maximization problem with a linear objective function c x: (1.1) maximize c x subject to x F, where c denotes a constant vector in the n-dimensional Euclidean space R n and F a subset of R n. We can reduce a more general maximization problem with a nonlinear objective function f(x) to a maximization problem having a linear objective function represented by a new variable, x n+1, if we replace f(x) by x n+1 and then add the inequality f(x) x n+1 to the constraint. hus (1.1) covers such a general optimization problem. hroughout the paper we assume that F is compact. hen the problem (1.1) has a global maximizer whenever the feasible region F is nonempty. For any compact convex set C containing F, the maximization problem (1.2) maximize c x subject to x C serves as a convex relaxation problem, which satisfies the properties that (i) the maximum objective value ζ of the problem (1.2) gives an upper bound for the maximum objective value ζ of the problem (1.1), i.e., ζ ζ, and Received by the editors March 31, 1998; accepted for publication (in revised form) July 19, 1999; published electronically March 21, Department of Mathematical and Computing Sciences, okyo Institute of echnology, Oh-Okayama, Meguro-ku, okyo , Japan (kojima@is.titech.ac.jp). Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada (ltuncel@math.uwaterloo.ca). his work was completed while this author was visiting okyo Institute of echnology, Department of Mathematical and Computing Sciences, on a sabbatical leave from University of Waterloo. he research of this author was supported in part by okyo Institute of echnology and by a research grant from NSERC of Canada. 750

2 SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES 751 (ii) if a maximizer ˆx C of (1.2) lies in F, it is a maximizer of (1.1). Since the objective function of (1.1) is linear, we know that if we take the convex hull c.hull(f) (defined as the intersection of all the convex sets containing F) for C in (1.2), then (i) ζ = ζ, and (ii) the set of the maximizers of (1.2) forms a compact convex set whose extreme points are maximizers of (1.1). herefore, if we solve the relaxation problem (1.2) with a convex feasible region C which closely approximates c.hull(f), we can expect to get not only a good upper bound ζ for the maximum objective value ζ but also an approximate maximizer of the problem (1.1). We can further prove that for almost every c R n (in the sense of measure), any maximizer x C = c.hull(f) of (1.2) is an extreme point of c.hull(f), which also lies in F; hence x is a maximizer of (1.1). his follows from a result due to Ewald, Larman, and Rogers [5] for consequences of related results; see also [17]. Furthermore, for many representations of various convex sets C, given ˆx C, we can very efficiently find x, an extreme point of C, such that c x c ˆx. Indeed, the relaxation technique mentioned above has been playing an essential role in practical computational methods for solving various problems in the fields of combinatorial optimization and global optimization. It is often used in hybrid schemes with the branch-and-bound and branch-and-cut techniques in those fields. See, for instance, [2]. he aim of this paper is to present a basic idea on how we can approximate the convex hull of F. his is a quite difficult problem, and also too general. Before making further discussions, we at least need to provide an appropriate (algebraic) representation for the compact feasible region F of the problem (1.1) and the compact convex feasible region C of the relaxation problem (1.2). We employ quadratic inequalities for this purpose. Let S n and S+ n S n denote the set of n n symmetric matrices and the set of n n symmetric positive semidefinite matrices, respectively. Given Q S n, q R n, and γ R, we write a quadratic function on R n with the quadratic term x Qx, the linear term 2q x, and the constant term γ as p( ; γ, q, Q): p(x; γ, q, Q) γ + 2q x + x Qx x R n. hen the set Q of quadratic functions on R n and the set Q + of convex quadratic functions are defined as and Q {p( ; γ, q, Q) : Q S n, q R n and γ R} Q + {p( ; γ, q, Q) : Q S n +, q R n and γ R}, respectively. We also write p( ) Q (or Q + ) instead of p( ; γ, q, Q) Q (or Q + ) if Q S n, q R n, and γ R are irrelevant. hroughout the paper, we assume that the feasible region F of the problem (1.1) is represented by a set of quadratic inequalities such that F = {x R n : p(x) 0 p( ) P F }, where P F denotes a set of quadratic functions, i.e., P F Q, and we will derive convex relaxations, C, represented by convex quadratic inequalities such that C = {x R n : p(x) 0 p( ) P C },

3 752 MASAKAZU KOJIMA AND LEVEN UNÇEL where P C denotes a set of convex quadratic functions, i.e., P C Q +. We allow cases where P F and/or P C involve infinitely many quadratic functions. hus (1.1) or (1.2) (or both) can be a semi-infinite quadratic optimization problem (QOP). Here we use the word semi-infinite for optimization problems having a finite number of scalar variables and possibly an infinite number of inequality constraints. here are some reasons why we have chosen quadratic inequalities for the representation of both problems, the maximization problem (1.1) that we want to solve and its convex relaxation problem (1.2). First, quadratic inequalities form a class of relatively easily manageable nonlinear inequalities, yet they have enough power to describe any compact feasible region F in R n. Indeed, if F is closed, then its complement R n \F is open so that it can be represented as the union of the open balls {x R n : (x x ) (x x ) < ǫ(x )} with ǫ(x ) > 0 over all x G for some G R n \F; hence F = {x R n : (x x ) (x x ) ǫ(x ) x G}. We also know that any single polynomial inequality can be converted into a system of quadratic inequalities; for example, can be converted into x 2 1x 2 + 2x x 3 x 2 0, x 3 + x 2 0 and x 3 + 2x 2 x See [23, 24]. Second, we know that we can solve some classes of maximization problems having linear objective functions and a convex-quadratic-inequality constrained feasible region C efficiently. Among others, we can apply interior-point methods [1, 16] to the problem (1.2) when either P C is finite or P C is infinite, but its feasible region C is described as the projection of a set characterized by linear matrix inequalities in the space S n of n n symmetric matrices onto the n-dimensional Euclidean space R n. hird, and also most importantly, we can apply the semidefinite programming (SDP) relaxation, which was originally developed for 0-1 integer programming problems by Lovász and Schrijver [12] and later extended to nonconvex quadratic optimization problems [6, 18, 19], to the entire class of maximization problems having a linear objective function and finitely or infinitely many quadratic inequality constraints. See also [1, 8, 9, 13, 15, 23, 24, 29]. In addition to the reasons above, we should mention that the maximization problem with a linear objective function and quadratic inequality constraints involves various optimization problems such as 0-1 integer linear (or quadratic) programming problems which, in principle, include all combinatorial optimization problems [1, 9, 18]. Linear complementarity problems [4], bimatrix games, and bilinear matrix inequalities [14, 20] are also included as special cases. For some optimization problems, some of the semidefinite programming (SDP) relaxations we provide may be solved in polynomially many iterations (of an interiorpoint method or an ellipsoid algorithm) approximately. Such conclusion requires, in the case of the ellipsoid method, the existence of a certain polynomial-time separation oracle for the underlying convex cone constraint (see [9]). In the case of interiorpoint algorithms (whose efficiency in the theory and practice of SDP has been well

4 SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES 753 established), we need to have an efficiently computable self-concordant barrier for the feasible solutions set or at least for the underlying cone constraints (see [16]). Some of the most exciting activities in combinatorial optimization are currently centered around the applications of SDP to combinatorial optimization problems (see [7]). Such activity in theory and practice is fueled by theoretical results establishing that certain simple SDP relaxations of a combinatorial optimization problem can be effectively utilized in developing polynomial-time approximation algorithms with worst-case approximation-ratio guarantees much better than those previously proven using linear programming or other techniques. (See Goemans [7], Goemans and Williamson [8], Nesterov [15], and Ye [29].) Also outstanding are the results on the stable set problem establishing the fact that SDP techniques can be used in optimizing over a relaxation of the stable set polytope which is contained in the polytope defined by the clique inequalities. (Note that it is NP-hard to optimize over the latter-mentioned polytope, whereas Grötschel, Lovász, and Schrijver [9] and Lovász, and Schrijver [12] were able to utilize polynomial-time methods to achieve a better goal, as far as the proof of approximate optimality of some feasible solutions of the stable set problem is concerned.) Given an initial approximation C 0 of F, i.e., a compact convex set C 0 containing F, both of the methods, proposed in this paper, generate a sequence of compact convex subsets C k (k = 1,2,...) of R n such that (a) c.hull(f) C k+1 C k (monotonicity), (b) k=1 C k = c.hull(f) (asymptotic convergence). It should be noted that the compactness of each C k and property (b) imply that (c) if F =, then k k=1 C k = for some finite number k (detecting infeasibility). o generate C k+1 at each iteration, the SDP relaxation and the linear programming (LP) relaxation play an essential role, and the entire method may be regarded as an extension of the Lovász Schrijver lift-and-project procedure for 0-1 integer programming problems to semi-infinite nonconvex quadratic optimization problems, with the use of the SDP relaxation in the first method and the LP relaxation in the second method. he LP relaxation, referred to above, is essentially the same as the reformulation-linearization technique developed for nonconvex quadratic optimization problems by Sherali and Alameddine [21]; see also [2, 22]. However, we should caution the reader that the methods presented here are mostly conceptual in the general settings, because we need to solve a semi-infinite SDP (or a semi-infinite LP) at each iteration. For such a task, an efficient practical algorithm may not be currently available. In their paper [6], Fujie and Kojima proposed the semi-infinite convex QOP relaxation for nonconvex quadratic optimization problems and showed that the semiinfinite convex QOP relaxation is not stronger than the SDP relaxation in general, but the two relaxations are essentially equivalent under Slater s constraint qualification. We establish the exact equivalence between the two relaxations for semi-infinite nonconvex quadratic optimization problems without any constraint qualification. Using this equivalence, we derive some fundamental features of our methods including (a) and (b) above. One of the common themes in this paper is the usage of cones of matrices (and duality) in our constructions. his was also one of the themes of [12]. he other themes of this paper are the successive applications of SDP relaxations and LP relaxations. We call the related procedures the successive SDP relaxation method and the successive semi-infinite LP relaxation method, respectively. Section 2 is devoted to preliminaries, where we provide some basic definitions

5 754 MASAKAZU KOJIMA AND LEVEN UNÇEL and properties on quadratic inequality representations for closed subsets of R n, the homogeneous form of quadratic functions, the SDP relaxation, etc. In section 3, we present our first method in detail as well as the main results, including the features (a) and (b). After we present some fundamental characterizations of the SDP relaxation in section 4, we give proofs of the main results in section 5. In section 6, we apply our method to 0-1 semi-infinite nonconvex quadratic optimization problems. Incorporating the basic results on the lift-and-project procedure given by Lovász and Schrijver [12] for 0-1 integer convex optimization problems, we show that our method terminates in at most (n + 1) iterations either to generate the convex hull of the feasible region or to detect the emptiness of the feasible region, where n denotes the number of 0-1 variables of the problem. Section 7 contains our second method, which is based on semi-infinite LP relaxations. We establish the same theoretical properties as we do for the successive SDP relaxation method. In section 8, we present two numerical examples showing the worst-case behavior of some of our procedures. In particular, we know from the second example that the best of our procedures requires infinitely many iterations to generate the convex hull of F in the worst case. 2. Preliminaries Semi-infinite quadratic inequality representation. In this subsection, we discuss some representations of a closed subset F of R n in terms of (possibly infinitely many) quadratic inequalities. If p( ; γ, q, Q) Q, and p(x; γ, q, Q) 0 holds for all x F, we say that p(x; γ, q, Q) 0 is a quadratic valid inequality for F and that p( ; γ, q, Q) induces a quadratic valid inequality for F. A quadratic valid inequality p(x; γ, q, Q) 0 for F is linear if Q = O, rank-1 quadratic if p(x) = (a x α)(a x β) for a R n, α R and β R such that α a x β x F, rank-2 quadratic if p(x) = (a x α)(b x β) for a R n, b R n, α R and β R such that a x α and b x β x F, spherical if p(x) = (x d) (x d) ρ for d R n and ρ > 0, ellipsoidal if p(x) = (x d) Q(x d) ρ for Q S n ++, d R n and ρ > 0, convex quadratic if Q S n +, respectively. It should be noted that if a quadratic valid inequality p(x; γ, q, Q) 0 for F is rank-2, then the rank of the matrix Q is at most 2 but that the converse is not necessarily true. We say that F has a (semi-infinite) quadratic inequality representation P Q if F = {x R n : p(x; γ, q, Q) 0 p( ; γ, q, Q) P} holds. o designate the underlying representation P of F, we often write F(P) instead of F. Whenever F is a closed proper subset of R n, F has infinitely many representations. We allow the cases where P consists of infinitely many quadratic functions. Hence p(x) 0 p( ) P can be a semi-infinite system of quadratic inequalities. If P Q is a quadratic inequality representation of F and if p( ) c.cone(p), then p(x) 0 is a quadratic valid inequality, where c.cone(p) denotes the closed convex cone generated by P. Hence if P P c.cone(p), then P is a quadratic inequality representation of F; F(P) = F(P ) = F(c.cone(P)). A quadratic inequality representation P of F is finite if it consists of a finite number of quadratic functions, and infinite otherwise. If F is a compact convex subset of R n, it has a quadratic inequality representation; in fact, the set of all the linear (rank-2 quadratic or spherical)

6 SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES 755 valid inequalities for F forms an inequality representation of F. If, in addition, F is polyhedral, we can take a finite linear inequality representation. Let C be a compact subset of R n. We use the following symbols: P L (C) = the set of p( ) s that induce linear valid inequalities for C, P 1 (C) = the set of p( ) s that induce rank-1 quadratic valid inequalities for C, P 2 (C) = the set of p( ) s that induce rank-2 quadratic valid inequalities for C, P S (C) = the set of p( ) s that induce spherical valid inequalities for C, P E (C) = the set of p( ) s that induce ellipsoidal valid inequalities for C, P C (C) = the set of p( ) s that induce convex quadratic valid inequalities for C, P (C) = the set of p( ) s that induce all quadratic valid inequalities for C. By definition, we see that ( P L (C) P 1 (C) P S (C) P E (C) ) P C (C) P (C), P S (C) P E (C) and ( P L (C) P 1 (C) ) P 2 (C) P (C). Note that if C is convex, then the equality C = {x R n : p(x) 0 p( ) P} holds with each P = P L (C), P 1 (C), P 2 (C), P S (C), P E (C), P C (C), P (C). Among these, P (C) is the strongest quadratic inequality representation of C Homogeneous form of quadratic functions lifting to the space of symmetric matrices. We introduce a different description of quadratic functions, which we call the homogeneous form. his form leads us to a lifting of a quadratic function defined on the Euclidean space to the space of symmetric matrices and to the SDP relaxation (or to the semi-infinite LP relaxation in section 4.2). For every quadratic function p( ; γ, q, Q) Q, we connect the variable vector x R n to the (1 + n) (1 + n) rank-1 positive semidefinite matrix x xx = ( 1 x ) (1, x ) S 1+n + and the triplet of the constant γ R, q R n, and Q S n to the (1 + n) (1 + n) symmetric matrix γ q q Q S 1+n. hen we have the identity ( γ q p(x; γ, q, Q) = (1, x ) q Q ) 1 = x ( γ q q Q ) x xx hus, if P Q is a quadratic inequality representation of F, then { } γ q P : p( ; γ, q, Q) P q Q provides an equivalent representation of F; { F(P) = x R n : P x xx } 0 P P. x R n. Now we have two kinds of description for a quadratic function on R n : the usual form p( ; γ, q, Q) = γ + 2q x + x Qx and the homogeneous form introduced above.

7 756 MASAKAZU KOJIMA AND LEVEN UNÇEL he former is used in section 5, where we prove our main results, while the latter is suitable for the compact description of the SDP relaxation in section 2.3 and the proof of its equivalence to the semi-infinite convex QOP relaxation in section 4. We will use both forms in parallel, choosing whichever is convenient to us in a given situation. It should be noted that the correspondence γ q p( ; γ, q, Q) Q S 1+n q Q is not only one-to-one but also linear. o save notation, we identify the set Q of quadratic functions with the set S 1+n of (1 + n) (1 + n) symmetric matrices and any subset of Q with the corresponding subset of S 1+n. Specifically, we write P = ( γ q ) P whenever p( ; γ, q, Q) P and identify the set of (1 + n) (1 + n) q Q symmetric matrices { } γ q : γ R, q R n, Q S n q Q with the set Q of quadratic functions from R n to R SDP relaxation. Let P be a semi-infinite quadratic inequality representation of F: F(P) = {x R n : p(x) 0 p( ) P} { } = x R n : P x xx 0 P P. he SDP relaxation ˆF(P) of F(P) with the quadratic inequality representation P is given by ˆF(P) x X S Rn : n such that S 1+n + and γ + 2q x + Q X 0 p( ; γ, q, Q) P X S n such that S = x R n + 1+n and :. P 0 P P If x F(P) and P P, then X = xx satisfies that ( 1 = (1, x x) ) S+ 1+n and P 0. his implies that x ˆF(P) and F(P) ˆF(P). We also see that ˆF(P) is convex. Hence c.hull(f(p)) ˆF(P). he SDP relaxation was originally proposed for combinatorial optimization problems and 0-1 integer programming problems [12], and later extended to quadratic optimization problems. See [1, 6, 8, 9, 15, 19, 18, 23, 24, 29]. 3. Main results. Now we are ready to describe our method for approximating a quadratic-inequality-constrained compact feasible region F of the minimization problem (1.1). Before running the method, we need to fix a semi-infinite quadratic

8 SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES 757 inequality representation P F of F, and choose an initial approximation C 0 of the convex hull of F, i.e., a compact convex set which contains c.hull(f). Starting from C 0, the method generates a sequence of compact convex sets C k (k = 0,1,2,...), which we expect to converge to c.hull(f). At each iteration, we choose a semi-infinite quadratic inequality representation P k of the kth approximation C k of c.hull(f). Since c.hull(f) C k, the union (P F P k ) forms a semi-infinite quadratic inequality representation of F. We then apply the SDP relaxation to (P F P k ) to generate the next iterate C k+1 = ˆF(P F P k ). It should be emphasized that during none of the iterations do we modify or strengthen the representation P F directly. We only utilize the semi-infinite quadratic inequality representation of the compact convex set C k that has been computed in the previous iteration. Successive SDP Relaxation Method. Step 0: Let k = 0. Step 1: If C k = or C k = c.hull(f), then stop. Step 2: Choose a semi-infinite quadratic inequality representation P k for C k. Step 3: Let (3.1) C k+1 = ˆF(P F P k ) X S n such that = x R n : and P 0 S 1+n + P P F P k. Step 4: Let k = k + 1, and go to Step 1. We state two convergence theorems below. We choose the spherical inequality representation P S (C k ) for C k at Step 2 of each iteration in the first theorem, while we choose the rank-2 quadratic inequality representation P 2 (C k ) for C k at Step 2 of each iteration in the second theorem. heir proofs will be given in section 5. heorem 3.1. Assume that P F is a semi-infinite quadratic inequality representation of a compact subset F of R n, and that C 0 F is a compact convex subset of R n. If we choose P k = P S (C k ) at Step 2 of each iteration in the successive SDP relaxation method, then the monotonicity property (a) and the asymptotic convergence property (b) stated in the introduction hold. heorem 3.2. Under the same assumptions as in heorem 3.1, if we choose P k = P 2 (C k ) at Step 2 of each iteration in the successive SDP relaxation method, then (a) and (b) remain valid. We know that if P Q and P Q are semi-infinite quadratic inequality representations of C k and if P P, then ˆF(P ) ˆF(P). Hence, even if we replace P k = P S (C k ) in heorem 3.1 by P k P S (C k ) (or P k = P 2 (C k ) in heorem 3.2 by P k P 2 (C k ) ), the properties (a) and (b) remain valid. In particular, (a) and (b) remain valid when we choose any of P E (C k ), P C (C k ), and P (C k ) for P k. If we take the linear representation P L (C k ) of C k at every iteration, then we can prove that C 1 = F(P F P 0 ) = F(P F ) C 0 and C k+1 = F(P F ) C k = C 1 (k = 1,2,...). (See Lemma 4.1.) Hence (b) does not follow in general. In section 8, we will give two numerical examples. he first example shows that the rank-1 quadratic inequality representation P k = P 1 (C k ) is not strong enough

9 758 MASAKAZU KOJIMA AND LEVEN UNÇEL to ensure (b). he second example shows that even when we choose the strongest quadratic inequality representation P (C k ) of C k for P k at every iteration, not only does the convergence C k c.hull(f) require infinitely many iterations, but its speed also becomes extremely slow in the worst case. 4. Fundamental characterization of successive convex relaxation Semi-infinite convex QOP relaxation and its equivalence to SDP relaxation. he semi-infinite convex QOP relaxation of F(P) with the semi-infinite quadratic inequality representation P is defined as We observe that F(P) {x R n : p(x) 0 p( ) c.cone(p) Q + } { } = x R n : P x xx 0 P c.cone(p) Q +. F(P) = { } x R n : P x xx 0 P c.cone(p) F(P) and that the set F(P) is a closed convex set. Hence F(P) c.hull(f(p)) F(P). he semi-infinite convex QOP relaxation was introduced by Fujie and Kojima [6]. It was called the relaxation using convex-quadratic valid inequalities for F(P) in their paper [6]. he following basic properties of the relaxation are essentially due to them. Lemma 4.1. Let P F be a semi-infinite quadratic inequality representation of a closed set F R n. (i) Let P be a set of convex quadratic valid inequalities for F, i.e., P P C (F). hen F(P F P) F(P) = {x R n : p(x) 0 p( ) P}. (ii) Let P be a set of linear valid inequalities for F, i.e., P P L (F). hen F(P F P) = F(P F ) {x R n : p(x) 0 p( ) P}. (iii) Let x c.hull(f). Suppose that p(x ; γ, q, Q) 0 for some p( ; γ, q, Q) P F with a positive definite Q. hen x F(P F ). Proof. Part (i) follows directly from the definition of the semi-infinite convex QOP relaxation. Now we show (ii). Let C = {x R n : p(x) 0 p( ) P}. hen we see that F(P F P) F(P F ) F(P) = F(P F ) C. Hence it suffices to show that F(P F ) C F(P F P). Let p( ) c.cone(p F P) Q +. hen there exist p( ) i P F (i = 1,2,..., l), p( ) j P (j = l+1,..., m), and positive numbers λ i (i = 1,2,..., m) such that p( ) = l λ i p( ) i + m j=l+1 λ i p( ) i Q +. Since p( ) j P (j = l + 1,..., m) are linear functions, we see that l λ i p( ) i c.cone(p F ) Q + ; hence, l λ i p(x) i 0 x F(P F ).

10 SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES 759 Moreover, m j=l+1 λ i p( ) i c.cone(p) Q + ; hence, m j=l+1 λ i p(x) i 0 x C. herefore, p(x) = l λ i p i (x) + m j=l+1 λ i p i (x) 0 x F(P F ) C. his proves (ii). Finally we will show (iii). Since x F, there is a p ( ) P F such that p (x ) > 0. Hence, if ǫ > 0 is sufficiently small, we obtain that ǫp( ) + p( ) c.cone(p F ) Q + and ǫp (x ) + p(x ) > 0. his implies x F(P F ), and proves (iii). When P is finite and F(P) satisfies Slater s constraint qualification, Fujie and Kojima [6] showed that the semi-infinite convex QOP relaxation is essentially equivalent to the SDP relaxation in the sense that F(P) coincides with the closure of ˆF(P). he theorem below shows the exact equivalence between them, without any constraint qualification, for more general semi-infinite quadratic inequality representation cases. Since F(P) is closed, one of the consequences of the next theorem is that ˆF(P) is always closed. Note that we can assume without loss of generality that P is a closed convex cone, since every closed set F admits such a representation. heorem 4.2. Let P be a closed convex cone, giving a semi-infinite quadratic inequality representation of a closed subset F of R n ; F(P) = {x R n : p(x) 0 p( ) P}. hen its SDP relaxation and its semi-infinite convex QOP relaxation coincide with each other; ˆF(P) = F(P). Proof. Using the dual cone P = {V S : V U 0 U P} of P, we can express the sets ˆF(P) and F(P) as follows: { ˆF(P) = x R n : X S n such that } ( P ) S+ 1+n and { } F(P) = x R n : x xx (P Q + ) { [ ]} = x R n : x xx P S+ n. For the last identity above, we have used the fact that for any pair of closed convex cones K 1 and K 2 in R m, we have (K 1 K 2 ) = K 1 + K 2. First let x ˆF(P). hen there exists an X S n such that ( P ) S 1+n +.

11 760 MASAKAZU KOJIMA AND LEVEN UNÇEL Consider the identity [ ] 1 x 0 0 x xx = + x X 0 X xx. he first matrix on the right-hand side is in P and in the second matrix of the right-hand side, we have X xx S+ n since it is the Schur complement of 1 in the symmetric, positive semidefinite matrix. We have proved x F(P) and hence ˆF(P) F(P). For the converse, let x F(P); that is, there exists some H S n + such that x xx P. + H he matrix ( ) x xx + H is positive semidefinite if and only if (H + xx xx ) = H is. But the latter was already established. So, x xx ( P ) S 1+n + H +. herefore x ˆF(P), and F(P) ˆF(P) is proved Semi-infinite LP relaxation. In section 7, we will also need an analog of the above theorem for our successive semi-infinite LP relaxation method. For every semi-infinite quadratic inequality representation P of a compact subset F of R n, let us define ˆF L (P) { x R n : X S n such that P } 0 P P and F L (P) { x R n : γ + 2q x 0 p( ; γ, q, Q) c.cone(p) L } of Sherali and Alameddine [21]. Here, L denotes the set of linear functions on R n : L {p( ; γ, q, Q) Q : Q = O}. he next result can be obtained by following the steps of the proof of heorem 4.2. Corollary 4.3. Let P be a closed convex cone, giving a semi-infinite quadratic inequality representation of a closed subset F of R n ; F(P) = {x R n : p(x) 0 p( ) P}. hen ˆF L (P) = F L (P). Proof. We observe that { } ˆF L (P) = x R n : X S n such that P

12 SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES 761 and { } F L (P) = x R n : x xx (P L) { [ ]} = x R n : x xx P S n. Since it is easy to see that X S n such that P if and only if [ ] x xx P S n, the proof is complete Invariance under one-to-one affine transformation. Let f(x) = Ax+ b be an arbitrary one-to-one affine transformation on R n, where A is an n n nonsingular matrix and b R n. hen f( ˆF(P)) = f( F(P)) = {y R n : p (y) 0 p ( ) c.cone(p ) Q + }, f( ˆF L (P)) = f( F L (P)) = {y R n : p (y) 0 p ( ) c.cone(p ) L}, where P {p(f 1 ( )) : p( ) P} forms a semi-infinite quadratic inequality representation of f(f(p)). his means that the semi-infinite SDP and LP relaxations are invariant under the one-to-one affine transformation f(x) = Ax + b. We also see that P U (f(c)) = {p(f 1 ( )) : p( ) P U (C)} holds, where U {L, 1, 2, E, C, }. herefore, P L (C), P 1 (C), P 2 (C), P E (C), P C (C), and P (C) are invariant under one-to-one affine transformations on R n. If in addition A is a scalar multiple of an orthogonal matrix, then the above identity also holds for U = S; hence P S (C) is invariant under such a one-to-one affine transformation on R n. At each iteration of the successive SDP relaxation method, we observe that f(c k+1 ) = {y R n : p (y) 0 p ( ) c.cone(p F P k) Q + }, where P F {p(f 1 ( )) : p( ) P F } forms a semi-infinite quadratic inequality representation of f(f) and P k {p(f 1 ( )) : p( ) P k } forms a semi-infinite quadratic inequality representation of f(c k ). Furthermore, if we choose one of the invariant semi-infinite quadratic inequality representations P L (C k ), P 1 (C k ), P 2 (C k ), P E (C k ), P C (C k ), and P (C k ) of C k under any one-to-one affine transformation for P k, we see that P U (f(c)) = {p(f 1 ( )) : p( ) P U (C)}; hence the identity above turns out to be f(c k+1 ) = { y R n : p (y) 0 p ( ) c.cone(p F P U (f(c k ))) Q + }. Here U {L, 1, 2, E, C, }. herefore the successive SDP relaxation method is invariant under any one-to-one affine transformation. he same comment applies to the successive semi-infinite LP relaxation method, which we will present in section 7.

13 762 MASAKAZU KOJIMA AND LEVEN UNÇEL 5. Proofs of heorems 3.1 and 3.2. We present three lemmas, Lemma 5.1 in section 5.1, Lemma 5.2 in section 5.2, and Lemma 5.3 in section 5.4. Lemma 5.1 proves the monotonicity property (a) in heorems 3.1 and 3.2 simultaneously. Lemma 5.2 is used to prove heorem 3.1 in section 5.3, and Lemma 5.3 to prove heorem 3.2 in section Monotonicity. We first establish the monotonicity in general. Lemma 5.1. Let C 0 be a compact convex set containing F. Fix a closed convex cone S+ 1+n K S 1+n and U {L,1,2, S, E, C, }. Define X S n such that K and C k+1 x R n : P 0 P P F P U (C k ) for k = 1,2,.... Assume that X S n such that K and C 0 = x R n : P 0 P P U (C 0 ). hen c.hull(f) C k+1 C k for all k = 0,1,2,.... Proof. Since K S+ 1+n, it contains all symmetric rank-1 matrices of the form x xx. Now, as in the arguments in section 2.3, it follows that c.hull(f) C k for all k = 0,1,2,.... We will show by induction that C k+1 C k for all k = 0,1,.... By the construction of C 1 and the assumption imposed on C 0, we first observe that C 1 C 0. Now assume that C k C k 1 for some k 1. hen P U (C k 1 ) P U (C k ), which implies that P F P U (C k 1 ) P F P U (C k ). herefore, C k+1 C k, as desired Separating hypersphere. he following lemma easily follows from the separating hyperplane theorem, and the proof is omitted here. Lemma 5.2. Let C be a compact convex subset of R n and x C. hen there exists a hypersphere S {x R n : x d = η} which strictly separates the point x and C such that (5.1) x d > η > x d x C, where d R n and η > Proof of heorem 3.1. he monotonicity property (a) follows from Lemma 5.1 by letting K S+ 1+n and U S. Let C k=0 C k. We know by (a) that c.hull(f) C C k+1 C k (k = 0,1,...), and that all the sets c.hull(f), C, and C k are compact sets. o prove (b), we have the following left to show: C c.hull(f). Assume on the contrary that there exists some x C such that x c.hull(f). hen, by Lemma 5.2, there exists a hypersphere S {x R n : x d = η} that strictly separates the point x C from c.hull(f) such that x d > η > x d x c.hull(f),

14 SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES 763 where d R n and η > 0. Let η sup{ x d : x C}. Obviously, η < η = x d for some x C. Since x c.hull(f), there is a quadratic function, p 1 ( ; γ, q, Q) P F that cuts off x ; 0 < p 1 (x ; γ, q, Q). Note that if p 1 ( ; γ, q, Q) is such a quadratic function, then so is αp 1 ( ; γ, q, Q) for any α > 0. Hence we may assume that the minimum eigenvalue of the matrix Q S n is at least ( 1). Now consider a quadratic function p 2 ( ) defined by p 2 (x) = (x d) (x d) (η ) 2 p 1 (x ; γ, q, Q)/2 x R n. By the definition of η, we see that p 2 (x) p 1 (x )/2 < 0 x C. his means that the open ball B + {x R n : p 2 (x) < 0} with the center d and the radius (η ) 2 + p 1 (x ; γ, q, Q)/2 forms a neighborhood of the compact set C. On the other hand, the sequence {C k } of compact subsets of R n satisfies C k+1 C k (k = 0,1,2,...) and C = k=0c k. So, we can find a finite positive number l such that the open ball B + contains C l. Hence, p 2 (x) 0 is a convex quadratic valid inequality for C l ; p 2 ( ) P l. We also see that p 1 (x ; γ, q, Q) + p 2 (x ) = p 1 (x ; γ, q, Q)/2 > 0 and p 1 ( ; γ, q, Q) + p 2 ( ) Q +. hus we have shown that p 1 (x ; γ, q, Q) + p 2 (x ) > 0 and p 1 ( ; γ, q, Q) + p 2 ( ) c.cone(p F P l ) Q +. herefore, x C l+1 = F(P F P l ), so that x C = k=0 C k. his is a contradiction. he theorem is proved A family of inequalities of the convex cone of rank-2 quadratic valid inequalities for the unit ball. Let B denote the unit ball {x R n : x 1}. Let Q be an arbitrary n n symmetric matrix, and let u R n be an arbitrary vector on the boundary of B; u = 1. We will construct a family of quadratic valid inequalities, which lie in the convex cone of rank-2 quadratic valid inequalities, p θ (x) 0, with a parameter θ (0, π/8) for the unit ball B satisfying the properties (i), (ii), and (iii) listed in Lemma 5.3. We first apply the eigenvalue decomposition to the matrix Q S n. We may assume that the first m eigenvalues are nonnegative and the last n m eigenvalues are nonpositive for some nonnegative integer m n. hen we can write the matrix Q S n as Q = m µ j r j r j j=1 n j=m+1 µ j r j r j, where r j = 1 (j = 1,2,..., n) and µ j 0 (j = 1,2,..., n), r j (j = 1,2,..., n) denote eigenvectors of Q, which are orthogonal to each other, and µ j (j = 1,2,..., m) and µ j (j = m + 1,..., n) denote the eigenvalues corresponding to them.

15 764 MASAKAZU KOJIMA AND LEVEN UNÇEL For each θ (0, π/8), we define a j (θ) u cos θ + r j sin θ (j = 1,2,..., n), ā j (θ) u cos θ r j sin θ (j = 1,2,..., n), b j +r j, b j r j (j = 1,2,..., m), b j r j, b j +r j (j = m + 1,..., n), (5.2) α j (θ) max{a j (θ) x : x B} = a j (θ) (j = 1,2,..., n), ᾱ j (θ) max{ā j (θ) x : x B} = ā j (θ) (j = 1,2,..., n), β j max{b j x : x B} = b j = 1 (j = 1,2,..., n), β j max{ b j x : x B} = b j = 1 (j = 1,2,..., n), µ j λ j (θ) 0 (j = 1,2,..., n). 2 sin θ hen, θ (0, π/8) and j = 1,2,..., n, a j (θ), ā j (θ), b j (θ), and b j (θ) are nonzero vectors, and { a j (θ) x α j (θ) 0, b j x β j 0, (5.3) ā j (θ) x ᾱ j (θ) 0, b j x β j 0 are linear valid inequalities for the unit ball B. For all θ (0, π/8), define (5.4) p θ (x) n j=1 ( λ j (θ) (a j (θ) x α j (θ))(b j x β j ) + (ā j (θ) x ᾱ j (θ))( b j x β ) j ). hen p θ ( ) c.cone(p 2 (B)) for all θ (0, π/8). In particular, p θ (u) 0 θ (0, π/8). Lemma 5.3. (i) p θ ( ) c.cone(p 2 (B)). (ii) p θ (u) 0 as θ (0, π/8) tends to 0. (iii) he Hessian matrix of p θ ( ) coincides with Q. Proof. Part (i) was already shown. (ii) Let j be fixed. It suffices to show that ǫ j (θ) λ j (θ)(a j (θ) u α j (θ))(b j u β j ) and ǫ j (θ) λ j (θ)(ā j (θ) u ᾱ j (θ))( b j u β j ) converge to zero as θ (0, π/8) tends to 0. First, we derive that ǫ j (θ) converges to zero as θ (0, π/8) tends to 0. We see from (5.2) that (5.5) ǫ j (θ) = µ j(cos θ + u r j sin θ u cos θ + r j sin θ ) (b j u 1) 2 sin θ µ j (cos θ + u r j sin θ (cos 2 θ + 2u r j sin θ cos θ + sin 2 θ) 1 2 = 2 sin θ (b j u 1). ) Since both the numerator and the denominator above converge to zero as θ (0, π/8) tends to 0, we calculate their derivatives at θ = 0. he derivative of the numerator

16 SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES 765 turns out to be ( µ j sin θ + u r j cos θ + u r j (sin 2 θ cos 2 ) θ) (b (2u r j sin θ cos θ + 1) 1/2 j u 1), which vanishes at θ = 0. On the other hand, the derivative 2 cos θ of the denominator 2 sin θ in (5.5) does not vanish at θ = 0. hus, ǫ j (θ) converges to 0 as θ (0, π/8) tends to 0. Similarly, we can prove that ǫ j (θ) converges to 0 as θ (0, π/8) tends to 0. (iii) It follows from the definitions (5.2) and (5.4) that the Hessian matrix of the quadratic function p θ ( ) = = n j=1 λ j (θ) a j(θ)b j + b j a j (θ) + ā j(θ) b j + b j ā j (θ) 2 m µ j r j r j + j=1 = Q. n j=m+1 µ j r j r j From the lemma above, we see that the cone P 2 (B) is rich enough to contain rank-2 quadratic functions with any prescribed Hessian, leading to valid inequalities that are tight at any given point on the boundary of B Proof of heorem 3.2. he monotonicity property (a) follows from Lemma 5.1 by letting K S+ 1+n and U 2. o derive (b), it suffices to show that C k=0 C k c.hull(f) as in the proof of heorem 3.1. Assume on the contrary that x c.hull(f) for some x C. By Lemma 5.2, there exists a hypersphere S {x R n : x d = η} which strictly separates the point x C and c.hull(f) such that x d > δ > x d x c.hull(f), where d R n and δ > 0. Let δ sup{ x d : x C}. Obviously, δ = u d > δ for some u C. Since the successive SDP relaxation method using the rank-2 quadratic representation for C k at each iteration is invariant under the affine transformation (x d)/δ x, which maps d to the origin and the hypersphere S {x R n : x d = δ } onto the unit hypersphere {x R n : x = 1}, we may assume that d = 0 and δ = 1. hus, we have obtained that C B {x R n : x 1} and u C, u c.hull(f), u = 1. Since u F, there is a quadratic function p 1 ( ; γ, q, Q) P F that cuts off u; p 1 (u; γ, q, Q) > 0. Now, let p θ ( ) P 2 (B) Q + be the quadratic function introduced in section 5.4. See (5.2) and (5.4). By Lemma 5.3, we can choose a θ (0, π/8) for which p θ (u) p 1 (u; γ, q, Q)/3 holds. Now we define α k j = max{a j (θ) x : x C k }, β k j = max{b j (θ) x : x C k } (1 j n), ᾱj k = max{ā j (θ) x : x C k }, βk j = max{ b j (θ) x : x C k } (1 j n), n p k(x) = λ j (θ) ( (a j (θ) x αj)(b k j (θ) x βj k ) j=1 +(ā j (θ) x ᾱ k j)( b j (θ) x β k j ) )

17 766 MASAKAZU KOJIMA AND LEVEN UNÇEL for k = 0,1,2,.... By construction, we know that p k ( ) c.cone(p2 (C k )). Since both quadratic functions p θ ( ) and p k ( ) have the common Hessian matrix Q, p 1 ( ; γ, q, Q)+p k( ) c.cone(p F P k ) L c.cone(p F P k ) Q + k=0,1,2,.... We will show that (5.6) p 1 (u) + p k(u) p 1 (u)/3 > 0 for every sufficiently large k. hen the above two relations imply u C k+1 for such a large k. his contradicts the fact u C = k=0 C k. we see that Since the sequence of compact convex subsets C k (k = 0,1,2,...) satisfies C k+1 C k (k = 0,1,2,...) and k=0 C k = C B = {x : x 1}, α k j α j max{a j (θ) x : x C} α j (θ), β k j β j max{b j (θ) x : x C} β j, ᾱ k j ᾱ j max{ā j (θ) x : x C} ᾱ j (θ), βk j β j max{ b j (θ) x : x C} β j as k (j = 2,3,..., n). By continuity, we see then that for every sufficiently large k n p k(u) p 1 (u)/3 λ j (θ) ( (a j (θ) u αj)(b j (θ) u βj ) j=1 p 1 (u)/3 + p θ (u) 2p 1 (u)/3 (since p θ (u) p 1 (u)/3). +(ā j (θ) u ᾱ j)( b j (θ) u β j ) ) hus we have shown that (5.6) holds for every sufficiently large k. his completes the proof of heorem Application to 0-1 semi-infinite, nonconvex quadratic optimization problems. We briefly recall two of the Lovász Schrijver procedures for 0-1 integer programming problems, and relate them to our successive SDP relaxation method. Let F be a subset of {0,1} n whose convex hull is to be approximated. In the Lovász Schrijver procedures, we assume that a compact convex subset C 0 of R n satisfying F = C 0 {0,1} n is given in advance. We define K 0 {(λ, λx ) R 1+n : λ 0, and x C 0 }. Let K I denote the convex cone spanned by the 0-1 vectors in K 0 : K I = {(λ, λx ) R 1+n : λ 0, and x c.hull(f)}. Here the 0th coordinate is special. It is used in homogenizing the sets of interest in R n. Clearly C 0 = {x R n : (1, x ) K 0 } and c.hull(f) = {x R n : (1, x ) K I }. he closed convex cone K 0 serves as an initial relaxation of K I. Given the current relaxation K k of K I, first a convex cone M + (K k, K k ) in the space of (1 + n) (1 + n)

18 SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES 767 symmetric matrices is defined (the lifting operation). hen a projection of this cone gives the next relaxation N + (K k ) of K I. Now, we define the lifting operation in general. Let K and be closed convex cones in R 1+n. A (1 + n) (1 + n) symmetric matrix, Y, with real entries is in M + (K, ) if (i) Y S+ 1+n, (ii) Y e 0 = Diag(Y ), (iii) u Y v 0 u K, v. (his condition is equivalent to Y K.) Here, e 0 denotes the unit vector with 0th coordinate 1. Item (ii) above serves an important role in Lovász Schrijver procedures as well as in some of the SDP relaxations used by Goemans and Williamson [8], Nesterov [15], and Ye [29]. his equation is valid simply because for each j for which x j {0,1}, the equation x 2 j = x j is valid. Indeed, our general framework applies to any compact set in R n, and the equation Y e 0 = Diag(Y ) was not utilized in earlier sections (as it is not valid). In this section, however, the equation is valid and we utilize it. As will be noted in the proof of heorem 6.3, the inclusion of this equation will be guaranteed by our choice of the initial formulation. he third condition of Lovász Schrijver procedures is very interesting. hey present a couple of possibilities for the choice of cone in 0-1 integer programming. Among them is the cone spanned by all 0-1 vectors with the first component x 0 = 1. his choice, since the cone has a very simple set of generators, allows for the development of polynomial-time algorithms for approximately solving the successive SDP relaxations as long as the number of iterations of the successive procedure is O(1). heir result only assumes that a polynomial-time weak separation oracle is available for K. he key is that since has only O(n) extreme rays, it becomes trivial to check condition (iii) in polynomial time. On the other hand, Lovász and Schrijver [12] note that the choice K is also possible and leads to at least as good relaxations as the former choice for. (In many cases the successive relaxations for K are significantly tighter than the successive relaxations with the simpler choice of.) In the case of the latter choice, the possibility of polynomial-time solvability of the first few successive relaxations depends on the availability of polynomial-time algorithms to check Y K K. Our procedure uses K. Now, we describe the projection step. N + (K) {Y e 0 : Y M + (K, K)}. We also define the iterated operators N+(K) k as follows: N+(K) 0 := K and N+(K) k := N + (N+ k 1 (K)) for all integers k 1. (We use the notation N + (K), whereas N + (K, K) is used in [12].) Another procedure studied in [12] uses a weaker relaxation by removing the condition (i) in the lifting procedure. Let M(K, K) and N(K) denote the related sets for this procedure. We will refer to the first procedure using the lifting M + (K, K) (and the projection N + ) as the N + procedure. We will call the other (using M(K, K), and N) the N procedure. Lovász and Schrijver prove the following. heorem 6.1. and K N + (K) N 2 +(K) N n +(K) = K I K N(K) N 2 (K) N n (K) = K I.

19 768 MASAKAZU KOJIMA AND LEVEN UNÇEL Let us see how our successive SDP relaxation method applies to 0-1 nonconvex quadratic optimization problems. Consider a 0-1 nonconvex quadratic program: (6.1) minimize c x subject to x F {x {0,1} n : p(x) 0 p( ) P }. We may assume that the set P contains the quadratic functions x i (x i 1), i = 1,2,..., n. hen we can replace the 0-1 constraint imposed on the variable x i by the inequality x i (x i 1) 0. hus by adding the quadratic functions x i (x i 1), i = 1,2,..., n, to P, we obtain a quadratic inequality representation P F of the feasible region F. Let C 0 [0,1] n. Note that F C 0 {0,1} n = {0,1} n in our general setting here. However, F = C 0 {0,1} n has been assumed for some compact convex subset C 0 of R n in the Lovász Schrijver procedures discussed above. Lemma 6.2. Suppose that we take C 0 = [0,1] n and P 0 {x i (x i 1) : i = 1,2,..., n} P 2 (C 0 ). hen F = C 1 {0,1} n, where C 1 = x Rn : X S n such that Y = P Y 0 P P F P 0 S 1+n + and Proof. Let C 1 be the semi-infinite convex QOP relaxation of the set F with the quadratic inequality representation P F P 0 :. C 1 {x R n : p(x; γ, q, Q) 0 p( ; γ, q, Q) c.cone (P F P 0 ) Q + }. In view of heorem 4.2 and Lemma 5.1, we know that Hence it suffices to show that F c.hull(f) C 1 = C 1 C 0. {x C 1 : x i = 0 or 1, i = 1,2,..., n} F. If F contains all the 0-1 vectors, the inclusion relation above obviously holds. Now assume that x F is a 0-1 vector. hen there is a quadratic function p 1 (, γ, q, Q) P F such that p 1 (x, γ, q, Q) > 0. On the other hand, we know that the quadratic function p 2 (x) n x i (x i 1), with the identity matrix as its Hessian matrix, is a member of c.cone(p 0 ), and that p 2 (x ) = 0. Hence if ǫ > 0 is sufficiently small, then his implies that x C 1. ǫp 1 (, γ, q, Q) + p 2 ( ) c.cone (P F P 0 ) Q +, ǫp 1 (x, γ, q, Q) + p 2 (x ) > 0.

20 SUCCESSIVE CONVEX RELAXAIONS OF NONCONVEX SES 769 As a consequence of the lemma above, we see that the 0-1 nonconvex quadratic optimization problem (6.1) is equivalent to the 0-1 convex quadratic optimization problem (6.2) minimize c x subject to x F = C 1 {0,1} n. Using this observation, we can prove that in the case of 0-1 nonconvex quadratic optimization problem (6.1), our successive SDP relaxation method converges in (1+n) iterations. heorem 6.3. he successive SDP relaxation method, applied to the 0-1 nonconvex quadratic optimization problem (6.1), using C 0 = [0,1] n as the initial approximation of c.hull(f) and P k = P 2 (C k ) in each iteration, terminates in at most (1+n) iterations with C 1+n = c.hull(f). Proof. We note that by Lemma 6.2, after one iteration of the successive SDP relaxation method, we obtain the 0-1 convex quadratic optimization problem (6.2) that can be used with the original Lovász Schrijver procedure. We only have to note that the successive SDP relaxation method becomes the Lovász Schrijver procedure after the first iteration. For this purpose, we compare conditions (i), (ii), and (iii) of the Lovász Schrijver procedure for K = = K k to the conditions used to construct C k+1 = ˆF(P F P k ) in the successive SDP relaxation method. Here K k {(λ, λx ) R 1+n : λ 0, x C k }. First, we observe that X S n such that Y = S 1+n + if and only if λ 0, X S n such that Y = λ λx λ S 1+n +. Hence (i) is satisfied. For (ii), note that x i (x i 1) P F i implies the constraint Y e 0 Diag(Y ) and x i (x i 1) P F i implies Y e 0 Diag(Y ). Finally, for (iii), note that a linear inequality a x α is valid for C k if and only if (α, a ) Kk (recall C k = {x R n : (1, x ) K k }). herefore, we see that P 2 (C k ) = c.cone{ uv : u, v K k}. Step (3.1) of the successive SDP relaxation method implies that Y (P 2 (C k )). hus, we conclude by noting that = Y (P 2 (C k )) if and only if Y uv = u Y v 0 u, v K. Now, heorem 6.1 implies that n more steps of the procedure is sufficient. he above discussion and the results show that our successive SDP relaxation method generalizes the Lovász Schrijver N + procedure by ignoring condition (ii), which is no longer valid. Our results in the previous sections already showed that in this full generality, we still have the asymptotic convergence of the method. It is therefore interesting to investigate the same questions about the weaker procedure N: What is the generalization of procedure N? Does the generalization of procedure N satisfy the same theoretical properties as the successive SDP relaxation method? We answer both of these questions in the next section. As is shown in [12], in some cases the procedure N + is significantly better than N. Procedure N is weaker, but the relaxations given by it are always polyhedral sets (so LP techniques can be employed) and N + requires more general techniques. Hence, sometimes procedure N might be more manageable even if the procedure N + is not.